Problem: Simplify the following expression and state the conditions under which the simplification is valid. You can assume that $p \neq 0$. $a = \dfrac{-5p + 30}{p + 8} \times \dfrac{p^2 + 10p + 16}{p^2 + 2p} $
First factor the quadratic. $a = \dfrac{-5p + 30}{p + 8} \times \dfrac{(p + 8)(p + 2)}{p^2 + 2p} $ Then factor out any other terms. $a = \dfrac{-5(p - 6)}{p + 8} \times \dfrac{(p + 8)(p + 2)}{p(p + 2)} $ Then multiply the two numerators and multiply the two denominators. $a = \dfrac{ -5(p - 6) \times (p + 8)(p + 2) } { (p + 8) \times p(p + 2) } $ $a = \dfrac{ -5(p - 6)(p + 8)(p + 2)}{ p(p + 8)(p + 2)} $ Notice that $(p + 2)$ and $(p + 8)$ appear in both the numerator and denominator so we can cancel them. $a = \dfrac{ -5(p - 6)\cancel{(p + 8)}(p + 2)}{ p\cancel{(p + 8)}(p + 2)} $ We are dividing by $p + 8$ , so $p + 8 \neq 0$ Therefore, $p \neq -8$ $a = \dfrac{ -5(p - 6)\cancel{(p + 8)}\cancel{(p + 2)}}{ p\cancel{(p + 8)}\cancel{(p + 2)}} $ We are dividing by $p + 2$ , so $p + 2 \neq 0$ Therefore, $p \neq -2$ $a = \dfrac{-5(p - 6)}{p} ; \space p \neq -8 ; \space p \neq -2 $